Wednesday, March 21, 2007

A very brief tutorial on QM and SR

I write to Roy Simpson (who has an Oxford D.Phil. in the subject under Prof. Roger Penrose).

"... One day I will know enough QM to understand why it isn't mathematically secure! I currently even struggle to understand whether/how the Hilbert space setting of QM handles special relativity conceptually (in the sense of Minkowskian space-time)!"

Roy casually helps me out ... (may be useful to other perplexed individuals out there):

"QM on its own could be mathematically secure.

But QM + Special Relativity = unsecure (or at least uncertain as below)

Remember the Schrodinger equation itself:

H psi = d/dt psi

Here the H (Hamiltonian) might or might not be relativistic, but there is a lone d/dt on the RHS. This singles out "time" for special treatment in the Schrodinger equation different from space (d/dx terms). Any d/dx will be inside the Hamiltonian unrelated to this d/dt. So this is the root of the Newtonian aspect.

Meanwhile the psi lives in a Hilbert space.

Relativity/Minkowski space would prefer a term like: d/dx + d/dy + d/dz - d/dt,even better with these components squared. When this is done we end up with the Dirac equation for a specific type of particle like the electron. Elegant but not the original Schrodinger equation which is somehow also meant to be valid even for a relativistic electron. The Schrodinger aspect gets ignored in physical practice as all the numbers now come from the Dirac equation. So if you ask too many questions in the relativistic case about the Schrodinger equation you end up in inconsistency. Schrodinger himself kind of knew this, but opted for his equation as it handled the non-relativistic case well - and the relativistic equations came later.

When I raised the query at the top of the page, I was thinking of a Minkowskian metric over the Hilbert space. Wrong! The Hilbert space is an abstract space, not space-time. In fact, space-time is just a particular basis out of many, corresponding to event-positional observables (although, as Roy pointed out, time is 'special').

In fact, the way Special Relativity comes into Quantum Mechanics is by way of Quantum Field Theory, as described in 'Deep Down Things' which I reviewed here.

I find QM and SR one vast jigsaw puzzle and the pieces are only slowly coming together as I study more.